- A manifold with an affine connection is called a locally symmetric affine space if the torsion tensor and the covariant derivative of the curvature tensor vanish identically.
- A (pseudo-) Riemannian manifold is called a locally symmetric (pseudo-) Riemannian space if the covariant derivative of its curvature tensor with respect to the Levi-Civita connection vanishes identically.
- A pseudo-Riemannian manifold (respectively, a manifold with an affine connection) $M$ is called a globally symmetric pseudo-Riemannian (affine) space if one can assign to every point $x \in M$ an isometry (affine transformation) $S_x$ of $M$ such that $S_x^2 = id$ and $x$ is an isolated fixed point of $S_x$.
- Let $G$ be a connected Lie group, let $\Phi$ be an involutive automorphism (i.e. $\Phi^2 = id$), let $G^\Phi$ be the closed subgroup of all $\Phi$-fixed points, let $G_0^\Phi$ be the component of the identity in $G^\Phi$, and let $H$ be a closed subgroup of $G$ such that$$G_0^\Phi \subset H \subset G^\Phi$$Then the homogeneous space $G/H$ is called a symmetric homogeneous space.
- A symmetric space in the sense of Loos (a Loos symmetric space) is a manifold $M$ endowed with a multiplication$$M \times M \longrightarrow M, \qquad (x,y) \mapsto x.y$$satisfying the following four conditions:
- every point $x \in M$ has a neighbourhood $U$ such that $x.y=y$ implies $y=x$ for all $y \in U$.
- the covariant derivative of the curvature tensor vanishes;
- every geodesic $\gamma$ is a trajectory of some one-parameter subgroup $\psi$ of $G$, and parallel translation of vectors along $\gamma$ coincides with their translation by means of $\psi$; and
- the geodesics are closed under multiplication (they are called one-dimensional subspaces).
|||P.A. Shirokov, "Selected works on geometry" , Kazan' (1966) (In Russian)|
|[2a]||E. Cartan, "Sur une classe rémarkable d'espaces de Riemann" Bull. Soc. Math. France , 54 (1926) pp. 214–264|
|[2b]||E. Cartan, "Sur une classe rémarkable d'espaces de Riemann" Bull. Soc. Math. France , 55 (1927) pp. 114–134|
|||M. Berger, "Les espaces symmétriques noncompacts" Ann. Sci. École Norm. Sup. , 74 (1957) pp. 85–177|
|||O. Loos, "Symmetric spaces" , 1–2 , Benjamin (1969)|
|||S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)|
Let be a globally symmetric Riemannian space, the connected component of the group of isometries of and the isotropy subgroup of of some point of . Then definitions can be given for being of compact, non-compact or Euclidean type in terms of the corresponding pair of Lie algebras . In particular, if is of the non-compact type, then has a Cartan decomposition , see .
|[a1]||A.L. Besse, "Einstein manifolds" , Springer (1987)|
|[a2]||B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)|
Symmetric space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Symmetric_space&oldid=28422